We will now probe a little into the corresponding negative integer values, attempting to properly explain their important role.

ζ(s) = 1/1

^{s }+ 1/2

^{s }+ 1/3

^{s}+ 1/4

^{s }+ .......

Therefore, when for example s = – 2,

ζ(– 2) = 1/1

^{– 2 }+ 1/2

^{– 2 }+ 1/3

^{– 2 }+ 1/4

^{– 2 }+......

= 1

^{ 2 }+ 2^{ 2 }+ 3^{ 2 }+ 4^{ 2 }+........ = 1 + 4 + 9 + 16 + ......
From the standard linear interpretation, this infinite series clearly diverges (with no finite result).

However in terms of the Riemann zeta function ζ(– 2) = 0 (in what is referred to as the first of the trivial zeros).

Now this choice of expression is somewhat unfortunate. Whereas it is true that the trivial zeros do not play a significant role with respect to the quantitative nature of the primes, they do indeed have an extremely important qualitative function (which is entirely overlooked in the conventional approach).

In fact I have seen anything resembling a satisfactory explanation in conventional terms as to why the series 1 + 4 + 9 + 16 + ...... can have two diametrically opposing values!

Whereas there is highly technical literature available on the nature of analytic continuation using holomorphic functions (on which the extension of the domain of the Riemann zeta function depends), this only helps to obscure the fact that no satisfactory explanation has yet been given as to why two opposing interpretations can exist for the same series!

It was my own determination to properly understand the nature of this first "trivial" zero that transformed my whole understanding of the Riemann zeta function.

I then slowly began to understand that the apparent nonsensical values of the function for values of s < 0, related to the fact that they do not correspond to a standard analytic interpretation (that is merely quantitative), but rather to an unrecognised holistic interpretation (that strictly is of a qualitative nature).

This in turn required that both of these aspects (analytic and holistic) must necessarily be viewed in a dynamic interactive context, where they are seen as directly complementary with each other.

So once again we cannot have number independence (in quantitative terms) without number interdependence (in a qualitative manner); likewise we cannot have number interdependence without number independence. so therefore in appropriate dynamic terms, both quantitative (analytic) and qualitative (holistic) aspects necessarily interact with each other in a bi-directional relative manner.

In particular this applies to interpretation of the Riemann zeta function which maps values for ζ(s) on the RHS of the functional domain with corresponding values for ζ(1 – s) on the LHS.

So therefore when we apply the standard analytic interpretation in quantitative terms to ζ(s) for s > 1, this implies that the corresponding interpretation for ζ(1 – s) should be carried out - relatively - in a complementary holistic manner.

Thus when ζ(3) with a recognised quantitative value in analytic terms, is mapped with the first of the trivial zeros i.e. ζ( – 2) this latter interpretation relates to a holistic - rather than analytic - value.

The clue to this holistic interpretation lies in the fact that one should now consider the dimensional power involved (i.e. – 2) in a qualitative rather than quantitative manner.

In standard analytic terms, 2 relates to units that are independent (as befits the cardinal notion of number); however from the qualitative perspective it relates to units that are interdependent with each other (that befits the ordinal notion). So such interdependence of units implies that their positions can be interchanged, with each, depending on relative context, potentially existing as 1st or 2nd!

Now we have seen that we can indirectly express this qualitative view of number in quantitative terms, through obtaining the two roots of 1. In this way the two "units" can be expressed as + 1 and

– 1 respectively (where the signs can interchange depending on context).

– 1 respectively (where the signs can interchange depending on context).

I have mentioned many times before that such 2-dimensional appreciation, in holistic terms, characterises our understanding that the two turns at a crossroads can be both left and right, or in mathematical terms, both + 1 and – 1 (depending on the direction from which the crossroads is approached).

So with 1-dimensional interpretation only one polar reference frame is used. Thus when one approaches the crossroads from either a N or S direction (considered independently) one can unambiguously denote a turn at the crossroads as L or R.

However, 2-dimensional interpretation requires the ability to simultaneously "see" from two opposite i.e. complementary polar reference frames. Therefore when one simultaneously views the approach to the crossroads from both N and S directions, then (circular) paradox is generated from a dualistic (i.e. 1-dimensional) perspective. So each turn is now interchangeable as both left and right (which seemingly confounds normal logic). In this sense, one implicitly recognises that left and right are purely relative (with no meaning independent of each other).

This is all deeply relevant in mathematical terms, which is likewise conditioned by fundamental polarities that necessarily interact with each other in dynamic fashion.

So for example we cannot have a mathematical object without a corresponding subjective interpretation. So numbers as independent objective entities strictly have no meaning apart from the subjective mental interpretations we place on them!

Likewise we cannot have a quantitative without a corresponding qualitative dimension to numbers. In other words, the independence of numbers in quantitative terms necessarily implies their corresponding interdependence in a qualitative manner (and vice versa).

So these are necessarily polar relative terms that ultimately have no meaning apart from each other.

Yet we have spent millennia now attempting to interpret numbers (especially the primes) as if they somehow possess an absolute objective identity. And quite simply this is an utterly mistaken approach!

Now with reference to the general expression a

Therefore, in dynamic interactive terms, when a is interpreted in an analytic (quantitative) manner, b in complementary terms is thereby interpreted in a holistic (qualitative) fashion. Likewise in reverse when a is interpreted in holistic terms, b is then interpreted in a - relative - analytic fashion.

So all numbers, in base and dimensional terms have both (quantitative) analytic and (qualitative) holistic interpretations depending on relative context.

Thus when we look at 2 in this qualitative dimensional sense it implies - like with our crossroads - the simultaneous recognition of two complementary reference frames for number (i.e. Type 1 and Type 2). Though 2 in Type 1 and Type 2 terms represents a numerical value that seems unambiguous when considered separately, deep paradox arises when the frames are incorporated simultaneously with each other.

In holistic terms, + is always associated with the conscious notion of positing phenomena; however, – carries the holistic meaning of negation (i.e. of making what is conscious, unconscious). And the unconscious element of understanding then expresses itself as intuition (i.e. a psycho spiritual energy) which entails direct appreciation of the mutual interdependence of phenomena.

So just like the combination of matter and anti-matter particles in physics leads to the direct generation of physical energy, likewise the combination of psychic matter and anti-matter objects, lead to the direct (unconscious) generation of psycho-spiritual energy, which is generally referred to as intuition!

Now whereas 2 represents the conscious attempt to portray the complementary nature of + 1 and

So with 1-dimensional interpretation only one polar reference frame is used. Thus when one approaches the crossroads from either a N or S direction (considered independently) one can unambiguously denote a turn at the crossroads as L or R.

However, 2-dimensional interpretation requires the ability to simultaneously "see" from two opposite i.e. complementary polar reference frames. Therefore when one simultaneously views the approach to the crossroads from both N and S directions, then (circular) paradox is generated from a dualistic (i.e. 1-dimensional) perspective. So each turn is now interchangeable as both left and right (which seemingly confounds normal logic). In this sense, one implicitly recognises that left and right are purely relative (with no meaning independent of each other).

This is all deeply relevant in mathematical terms, which is likewise conditioned by fundamental polarities that necessarily interact with each other in dynamic fashion.

So for example we cannot have a mathematical object without a corresponding subjective interpretation. So numbers as independent objective entities strictly have no meaning apart from the subjective mental interpretations we place on them!

Likewise we cannot have a quantitative without a corresponding qualitative dimension to numbers. In other words, the independence of numbers in quantitative terms necessarily implies their corresponding interdependence in a qualitative manner (and vice versa).

So these are necessarily polar relative terms that ultimately have no meaning apart from each other.

Yet we have spent millennia now attempting to interpret numbers (especially the primes) as if they somehow possess an absolute objective identity. And quite simply this is an utterly mistaken approach!

Now with reference to the general expression a

^{b}, both the base a and dimensional number b relate to differing reference frames that are quantitative and qualitative with respect to each other.Therefore, in dynamic interactive terms, when a is interpreted in an analytic (quantitative) manner, b in complementary terms is thereby interpreted in a holistic (qualitative) fashion. Likewise in reverse when a is interpreted in holistic terms, b is then interpreted in a - relative - analytic fashion.

So all numbers, in base and dimensional terms have both (quantitative) analytic and (qualitative) holistic interpretations depending on relative context.

Thus when we look at 2 in this qualitative dimensional sense it implies - like with our crossroads - the simultaneous recognition of two complementary reference frames for number (i.e. Type 1 and Type 2). Though 2 in Type 1 and Type 2 terms represents a numerical value that seems unambiguous when considered separately, deep paradox arises when the frames are incorporated simultaneously with each other.

In holistic terms, + is always associated with the conscious notion of positing phenomena; however, – carries the holistic meaning of negation (i.e. of making what is conscious, unconscious). And the unconscious element of understanding then expresses itself as intuition (i.e. a psycho spiritual energy) which entails direct appreciation of the mutual interdependence of phenomena.

So just like the combination of matter and anti-matter particles in physics leads to the direct generation of physical energy, likewise the combination of psychic matter and anti-matter objects, lead to the direct (unconscious) generation of psycho-spiritual energy, which is generally referred to as intuition!

Now whereas 2 represents the conscious attempt to portray the complementary nature of + 1 and